Question

How do you find the exact value of `cos(u-v)` given that `sinu=5/13` and `cosv=-3/5`?

Answer 1

`cos(u-v) = -16/65`

Explanation:

Find `cos u` using Pythagorean Theorem to find the adjacent side. Assume angle `u` is in the first quadrant:

1. `13^2 = 5^2 + a^2`
2. `a^2 = 169 - 25 = 144`
3. `a = 12`
   So `cos u = 12/13`

Find `sin v` using Pythagorean Theorem to find the adjacent side. Assume angle `v` is in the second quadrant:

1. `(-3)^2 + b^2 = 5^2`
2. `b^2 = 25 - 9 = 16`
3. `b = 4`
   So `sin v = 4/5`

Use the difference formula `cos(u-v) = cos u cos v + sin u sin v`:
`cos(u-v) = 12/13*-3/5 + 5/13*4/5 = -36/65 + 20/65 = -16/65`